Consider the matrix defined by block : $$ A = \begin{bmatrix} D_1 & B_{1,2} & \cdots & B_{1,n-1} \\ B_{1,2}^T & & \cdots & B_{2,n-1} \\ \vdots & \vdots & \ddots & \vdots \\ B_{1,n-1}^T & B_{2,n-2}^T & \cdots & D_n \end{bmatrix} $$ where $B_{i,j} = \Pi_{k=i}^{j-1} B_{k,k+1}$ (the $B$s are iterated products) and $D_i$ is diagonal.
Do matrix of this form have a name and are studied ?
Or maybe the simpler case of a matrix with the form $$ A = \begin{bmatrix} d_1 & b_{1,2} & \cdots & b_{1,n-1} \\ b_{1,2} & & \cdots & b_{2,n-1} \\ \vdots & \vdots & \ddots & \vdots \\ b_{1,n-1} & b_{2,n-2} & \cdots & d_n \end{bmatrix} $$ where $b_{i,j} = b_{i,i+1}\dots b_{j-1,j}$.